Search
Search Results
-
Würfel und Augensummen – ein unmögliches Paar
71-88Views:27It is well known that the values 2, 3, ..., 12 of the sum of eyes that appear when throwing two regular dice are not equally distributed. It can also be shown that no matter how the dice are falsified (or if only one of them is being manipulated) they can never reach the same probability concerning the sum of eyes ([8], 91 et seq.). This discovery can be generalized for n ≥ 2 dice. Various results of algebra and (real) calculus are used, so that a connection between two different mathematical fields can be realized. Such a connection is typical and often provides a large contribution for mathematics (because it frequently leads to a successful attempt of solving a special problem) and therefore examples of this sort should also be included in the mathematical education at schools as well as in the student teachers' university curriculum for the study of mathematics. -
Simple Variations on The Tower of Hanoi: A Study of Recurrences and Proofs by Induction
131-158Views:127The Tower of Hanoi problem was formulated in 1883 by mathematician Edouard Lucas. For over a century, this problem has become familiar to many of us in disciplines such as computer programming, algorithms, and discrete mathematics. Several variations to Lucas' original problem exist today, and interestingly some remain unsolved and continue to ignite research questions. Nevertheless, simple variations can still lead to interesting recurrences, which in turn are associated with exemplary proofs by induction. We explore this richness of the Tower of Hanoi beyond its classical setting to compliment the study of recurrences and proofs by induction, and clarify their pitfalls. Both topics are essential components of any typical introduction to algorithms or discrete mathematics.
Subject Classification: A20, C30, D40, D50, E50, M10, N70, P20, Q30, R20
-
Mathematical Doctoral School of the Mathematical Seminar of the University of Debrecen at the beginning of the 20th century (Debrecen, 1927-1940)
195-214Views:18In this article, we present the life and carrier of Professor Lajos Dávid, and those 16 mathematical dissertations, along with their authors, which were written under the supervision of Professor Dávid between 1927 and 1940. At the time mentioned, Lajos Dávid was the leader of the Mathematical Seminar of the University of Debrecen. The themes of the dissertations were connected with his scientific work, such as the history of mathematics (the two Bolyais), or his research work in mathematical analysis (arithmetic-geometric mean). -
Smartphones and QR-codes in education - a QR-code learning path for Boolean operations
111-120Views:36During the last few years new technologies have become more and more an integrative part of everyday life. The increase of the possession rate of smartphones by young people is especially impressive. This fact asks us educators to think about a didactically and pedagogically well designed integration of smartphones into our lessons and to bring in ideas and concepts. This paper describes a specific learning path where learners can work step by step on the topic Boolean Operations with QR-Code scanners which have been installed on their smartphones. Student teachers for mathematics who completed the learning path took part in a survey where they were asked questions about their willingness to integrate smartphones into their lessons. The results of the survey are presented in the second part of the paper. -
The role of computer in the process of solving of mathematical problems (results of research)
67-80Views:37We would like to present results of an almost two years investigations about the role computer in the process of solving of mathematical problems. In these investigations took part 35 students of the secondary school (generalists) in the age 17–19 years. Each of these students solved following problem:
Find all values of the parameter m so that the function
f(x) = |mx + 1| − |2x − m| is:
a) bounded,
b) bounded only from the bottom,
c) bounded only from above,
first without a computer and next with a special computer program. We would like to show results of these researches. -
Recalling calculus knowledge
55-70Views:34The main purpose of educational system is not only that the students perform well at the exam, but to remember the learnt material to some degree some time after the learning. This paper investigates students' retained knowledge, focusing mainly on topics concerning derivatives and differentiation, and examines the effect of re-learning in a short period of time. Results indicate that retained knowledge should be taken into consideration in instructional design and curriculum planning for the sequencing courses. -
Facilitating class attendance to improve student achievements
77-90Views:29Many studies have revealed that attendance is strongly associated with students' achievements, and have proposed different strategies to improve students' attendance. However, there are few studies investigating how to efficiently take students' attendance – the key component to improve students' attendance. Taking attendance manually is inefficient since it will consume part of the limited class time. This paper describes the design and the implementation of an online attendance system that is currently used in classes at West Virginia University and California University of Pennsylvania. Examples of the system are provided online. Implementation codes of the system are shared, which can be used to teach computer science courses such as Web Programming or Client-Server Script Languages. -
Pólya’s influence on (my) research
161-171Views:113In this article, I outline the influence of George Pólya's work on research in different areas and especially on mathematics education, namely heuristics and models of the problem-solving process. On a more personal note, I will go into some details regarding Pólya's influence on my own work in mathematical problem solving with a focus on the research project for my PhD thesis.
Subject Classification: 97xxx
-
On the psychology of mathematical problem solving by gifted students
289-301Views:35This paper examines the nature of mathematical problem solving from a psychological viewpoint as a sequence of mental steps. The scope is limited to solution processes for well defined problems, for instance, which occur at International Mathematical Olympiads. First the meta-mathematical background is outlined in order to present problem solving as a well defined search problem and hence as a discovery process. Solving problems is described as a sequence of elementary steps of the so called "relationship-vision" introduced here. Finally, non-procedural aspects of the psychology of problem solving are summarized, such as the role of persistence, teacher-pupil relationship, the amount of experience needed, self-confidence and inspiration at competitions. -
Eine geometrische Interpretation der Ausgleichsrechnung
159-173Views:27Using real examples of applied mathematics in upper secondary school one has do deal with inaccurate measures. This will lead to over constrained systems of linear equations. This paper shows an instructive approach which uses methods of descriptive and computer aided geometry to get a deeper insight into the area of calculus of observations. Using a qualified interpretation one can solve problems of calculus of observations with elementary construction techniques of descriptive geometry, independent of the norm one uses. -
Gaussian iteration of mean values and the existence of 2^(1/2)
35-42Views:34We propose a method for proving the existence of √2 and finding its approximate value in secondary education. -
Longest runs in coin tossing. Teaching recursive formulae, asymptotic theorems and computer simulations
261-274Views:39The coin tossing experiment is studied, focusing on higher education. The length of the longest head run can be studied by asymptotic theorems ([3]), by recursive formulae ([10]) or by computer simulations . In this work we make a comparative analysis of recursive formulas, asymptotic results and Monte Carlo simulation for education. We compare the distribution of the longest head run and that of the longest run (i.e. the longest pure heads or pure tails) studying fair coin events. We present a method that helps to understand the concepts and techniques mentioned in the title, which can be a useful didactic tool for colleagues teaching in higher education. -
Fehleranalyse beim Lösen von offenen Aufgaben Ergebnisse einer empirischen Studie in der Grundschule
83-113Views:12Open problems play a key role in mathematics education, also in primary school. However, children in primary school work in many relations in a different way from learner in secondary school. Therefore, the (possibly) first confrontation with an open task could be problematical. Within the framework of an international paper and pencil test it was examined how far children of primary school notice the openness of a task and which mistakes they do during working on that task. In particularly are meant by openness different interpretations of the task, which all lead to a set of numbers with more than one element as a result. For evaluation, a common classification system was adapted by slightly modification of the original system. -
WMI2: interactive mathematics on the web
393-405Views:15After 5 years of experiments and feedback we decided to continue the software development on WebMathematics Interactive, a web-based e-learning tool, rewriting it from scratch. The demonstration version of WebMathematics Interactive 2 (WMI2) has been shown to the expert audience on the CADGME conference. In this article we summarize the development goals and results. -
14 to 18-year-old Hungarian high-school students' view of mathematicians appearing in the media - a case study
183-194Views:10One way to develop positive attitude toward STEM subjects that popular media, including movies and films can be engaged to promote more positive and inclusive STEM images. The movie Hidden numbers offers an opportunity to explore the representations of scholars, especially mathematicians within a biographical drama. Focusing on 5 characters, this article first discusses whether these characters fit into stereotypical scientist image or not. Secondly, we examine how high school students evaluate these characters. We argue that this movie is suitable to promote positive attitude toward STEM subjects. -
Proof without words: partial sum and sum of a geometric series
423Views:4Let r be a positive real number such that 0 < r < 1, then:… -
The role of representations constructed by students in learning how to solve the transportation problem
129-148Views:107The purpose of the research presented in this paper was to study the role of concrete and table representations created by students in learning how to solve an optimization problem called the transportation problem. This topic was learned in collaborative groups using table representations suggested by teachers in 2021. In 2022, the researchers decided to enrich the students’ learning environment with concrete objects and urged the students to use them to present the problem to be solved. The students did it successfully and, to be able to record it in their notebooks, they constructed a table representation by themselves without any help from their teacher. After that, they managed to solve the problem by manipulating the objects. At the same time, each step in the solution was presented with changes in the table. The students were assessed before (pre-test) and after collaborative learning (test) in both academic years. The pre-test results were similar, but the test results were better in 2022. Therefore, it can be concluded that using concrete and table representations constructed by students in learning how to solve transportation problems makes collaborative learning more constructivist and more effective than when they use only table representations suggested by their teachers.
Subject Classification: 97M10, 97M40
-
Freudenthal fantasy on the bus, an American adaptation
133-142Views:62In the 1960’s two mathematicians, Hans Freudenthal in the Netherlands and Tamás Varga in Hungary, had argued that people learn mathematics by being actively involved and investigating realistic mathematical problems. Their method lives on in today’s teaching and learning through the various components of cooperative and active learning, by taking ownership in learning, and learning through student dialogue. The goal is to create a welcoming classroom atmosphere in which play takes the front seat. One such scenario is visiting various (animal) stations at the zoo by bus (illustrated by pictures). Passengers are getting on and off the bus at each station (illustrated by arrows), which is modeled on the open number line. This adapted and modified action research was carried out with 5-yearl-old children in public schools of Staten Island, NY in 2019.
Subject Classification: 97D40, 97F20, 97F30
-
The golden ratio and Viéte’s formula
43-54Views:11Viéte's formula uses an infinite product to express. In this paper we find a strikingly similar representation for the Golden Ratio. -
Illustrated analysis of Rule of Four using Maple
383-404Views:37Rule of Four, as a basic didactic principle, was formulated among the NCTM 2000 standards (see [14]) and since then it is quoted by numerous books and publications (see [4], [9], [12]). Practically we can say it is accepted by the community of didactic experts. The usage of the Rule of Four, however, has been realized mainly in the field of calculus, in fact certain authors restrict the wording of the principle to the calculus itself (e.g. [3]).
Calculus is a pleasant field, indeed. A sequence of values of a function provides us with example for numeric representation, while the formula and the graph of the function illustrate symbolic and graphical representations, respectively. In the end by wording the basic features of the function on natural language we gain textual representation.
This idyllic scene, however, becomes more complex when we leave the frame of calculus. In this paper we investigate the consequences of the usage of Rule of Four outside calculus. We discuss the different types of representations and show several examples which make the multiple features of representation evident. The examples are from different fields of mathematics and are created by the computer algebra system Maple, which turns out to be an excellent tool for illustration and visualization of the maim features of mathematical objects.
Next we introduce the concept of basic representation and rational representation, which is considered as the mathematical notion of "didactic usable" or "didactic rational" representation. In the end we generalize the notion of numeric representation, which leads us a more widely usable didactic principle which can be considered as a generalization of Rule of Four. -
Word problems in different textbooks at the early stage of teaching mathematics comparative analysis
31-49Views:151In a previous research, Csíkos and Szitányi (2019) studied teachers’ views and pedagogical content knowledge on the teaching of mathematical word problems. While doing so, they reviewed and compared Eastern European textbooks of Romania, Russia, Slovakia, Croatia, and Hungary to see how world problem-solving strategies are presented in commonly used textbooks. Their results suggested that teachers, in general, agreed with the approach of the textbooks regarding the explicit solution strategies and the types of word problems used for teaching problem-solving. They also revealed that the majority of the participants agreed that a word problem-solving algorithm should be introduced to the students as early as in the first school year. These results have been presented at the Varga 100 Conference in November 2019. As the findings suggested a remarkable similarity between the Eastern European textbook approaches, in the current study we decided to conduct further research involving more textbooks from China, Finland, and the United States.
Subject Classification: 97U20, 08A50
-
Zur Visualisierung des Satzes von Pythagoras
217-228Views:13In this article we make a study of a not-classical visualization of the theorem of Pythagoras using methods of elementary school geometry. We find collinear points, copoint straight lines and congruent pairs of parallelograms. The configuration of their midpoints induces a six-midpoint and a four-midpoint theorem. -
Equivalence and range of quadratic forms
123-129Views:10If two quadratic forms are equivalent, that is, if there is a linear transformation with integer coefficients and determinant 1 or −1 which takes one form to the other, then their ranges are the same and also their determinants are the same. The result of the paper is that for positive definite binary quadratic forms the converse is also true. Namely, if two positive definite binary quadratic forms of the same determinant have the same range, then they are equivalent. The arguments are guided by geometric considerations. -
The appearance of the characteristic features of the mathematical thinking in the thinking of a chess player
201-211Views:34It is more and more important in 21st century's education that not only facts and subject knowledge should be taught but also the ways and methods of thinking should be learnt by students. Thinking is a human specificity which is significant both in mathematics and chess. The exercises aimed at beginner chess players are appropriate to demonstrate to students the mathematical thinking of 12-14 year-old students.
Playing chess is an abstract activity. During the game we use abstract concepts (e.g. sacrifice, stalemate). When solving a chess problem we use logical quantifiers frequently (e.g. in the case of any move of white, black has a move that...). Among the endgames we find many examples (e.g. exceptional draw options) that state impossibility. Affirmation of existence is frequent in a mate position with many moves. We know there is a mate but the question in these cases is how it can be delivered.
We present the chess problem on beginners' level although these exercises appear in the game of advanced players and chess masters too, in a more complex form. We chose the mathematical tasks from arithmetic, number theory, geometry and the topic of equations. Students encounter these in classes, admission exams and student circles. Revealing the common features of mathematical and chess thinking shows how we can help the development of students' mathematical skills with the education of chess. -
Teaching integral transforms in secondary schools
241-260Views:34Today, Hungarian students in the secondary schools do not know the idea of complex numbers, and they can not integrate except those ones who learn mathematics in advance level. Without this knowledge we can teach Fourier transform for students. Why should we teach Fourier transform (FT) or Wavelet transform (WT) for them? To teach image file formats like JPEG, (JPEG2000) we need to talk about integral transforms. For students who are good in computer programming, writing the program of 1D FT or 2D FT is a nice task. In this article we demonstrate how we can teach Fourier and Wavelet transform for students in secondary school.